This is the sequel of a previous question.

Let us consider the sequence $$ \xi_n = 2n \{n\xi\}-n, $$ where $\xi>0$ is a given real irrational number and $\{\cdot\}$ is the fractional part.

Do there exist converging subsequences of $\{\xi_n\}_n$ to

finitereal numbers?

According to an answer, a necessary condition for the existence of cluster points is satisfied: the *normalized error*
$$
q^2 \left| \xi - \frac{p}{q} \right|
$$
is bounded with respect to $p$, $q \in \mathbb{N}$ and contains infinitely many different terms. However, it does not seem clear that the convergence of the normalized error along a subsequence entails the convergence of a subsequence of $\xi_n$, because $p$ and $q$ are not arbitrary integers. As far as I understand, diophantine approximation theory is not an obstruction. But is the answer affirmative? It is affirmative only for some class of irrational numbers?

More precisely, we have to produce (if any) a real number $x$ such that, for every $\epsilon>0$ we have for infinitely many integers $n$ $$ \left| n^2 \left(\xi - \frac{[n\xi]}{n}\right) - \frac{x+n}{2} \right| <\frac{\epsilon}{2}. $$